vektorfüggvény

A vektorfüggvény, or vector-valued function, is a function whose domain is a set of real numbers or a subset of real numbers, and whose codomain is a vector space. In simpler terms, it maps a real number (or multiple real numbers) to a vector.

The most common case involves a function from a single real variable to a vector in n-dimensional Euclidean space. For example, a function like r(t) = <x(t), y(t), z(t)> describes a curve in three-dimensional space, where t is the parameter (often representing time) and x(t), y(t), and z(t) are the scalar component functions. The domain of r(t) is a set of real numbers, and its codomain is R^3.

Vectorfüggvények are fundamental in various fields of mathematics and physics. They are used to describe paths of objects in motion, field lines in physics (like electric or magnetic fields), and parametric surfaces. The calculus of vectorfüggvények involves concepts like derivatives, which represent velocity or acceleration along a curve, and integrals, which can represent displacement or accumulated quantities.

When the domain of a vectorfüggvény is a subset of R^m and the codomain is R^n, it is sometimes referred to as a vector function of several variables. For instance, a function F(x, y) = <f(x, y), g(x, y)> maps points in a two-dimensional plane to vectors in R^2. These are crucial for understanding concepts like gradient fields and surface integrals.